(1) Field of the Invention
The present invention relates generally to computational fluid dynamics and more particularly to a method for determining the velocity of a three-dimensional fluid flow over a body submerged in the fluid.
(2) Description of the Prior Art
By determining the distribution (strength and location) of vorticity in a moving fluid, it is possible to determine the velocity and pressure fields in the fluid. Consequently, techniques which determine the strength and location of vorticity provide a powerful, flexible approach to computational fluid dynamics by focusing on an elemental aspect of fluid flow. The relationship equating vorticity to the curl of the velocity field, often introduced as the definition of vorticity, governs the generation of velocity from a prescribed distribution of vorticity. Appropriate boundary conditions complete the specification of the flow. The homogeneous solution to this equation, namely, velocity given by the gradient of a scalar function, is the familiar potential flow occurring exterior to regions of vorticity. For a fluid flow over a body, potential flows typically are invoked to investigate features for which the effects of the no-slip condition at the body surface are not crucial. These flows may be generated by velocity sources located on the body surface. The strength of these sources is determined by the no-flux condition at the surface. In the more general case, flow satisfying both the no-slip and the no-flux conditions at a body surface can be produced solely by a distribution of vorticity on the body surface.
An analogous relationship appears in magnetostatics, where Ampere's Law relates the curl of the magnetic field intensity to the current density. In this case, the current density is the source of the magnetic field intensity. Given a distribution of current density and relevant boundary conditions, the magnetic field intensity can be computed by the inversion of the differential equation. This inversion is, of course, the Biot-Savart Law. Similarly, for incompressible fluid flow this law provides the velocity field associated with a vorticity distribution.
This technique has advantages over the more widely pursued solution of the velocity-pressure form of the Navier-Stokes equations. One advantage is that the vorticity form of these equations does not contain pressure, unless the baroclinic term is retained. The absence of the pressure variable reduces the number of unknowns in the problem by one and allows the omission of the complex calculations required to determine the pressure. A second important advantage is that in many flows (e.g., boundary layers, wakes, jets, plumes, shear layers) the vorticity is confined to a relatively small portion of the space occupied by the flow. This concentration of the source of velocity allows a similar concentration of the computational resources on the region of significant vorticity.
The challenges presented in using vorticity to compute fluid flow are the accurate depiction of the spatial distribution of the vorticity field and the stepping of this field forward in time such that it obeys the vorticity equation. One prior art approach represents the vorticity field on a grid much in the same way that Navier-Stokes Solutions represent velocity and pressure on a grid. The grid points can be concentrated into the regions of the flow where vorticity is located. However, an accurate depiction of vorticity requires computation at numerous grid points.
A second prior art approach represents the vorticity field as a set of functions of compact support. The rate of change of location of members of the set is given by the local velocity and the strength of molecular diffusion. This latter approach is possible because vorticity is transported by the velocity field as material elements. This technique, known as the discrete-vortex-element method, does not have the problems of artificial viscosity often found in grid-based techniques and is therefore well-suited to cases of only slightly viscous (i.e., high Reynolds number) flow. The discrete-vortex-element technique is Lagrangian, so that for cases where part of the flow boundary moves with respect to other parts, the problem of adjusting a computational grid to the changing boundary locations is avoided. Finally, the technique amounts to a self-adapting grid, where the number density of elements tends to increase as local velocity gradients (vorticity) develop.
Since the ground-breaking work of Chorin, the discrete-vortex-element method has enjoyed substantial success in simulating two-dimensional flows. In particular, Chorin implements a vortex `blob`, where the vorticity is distributed uniformly over a disk in his two-dimensional calculations. Later, Chorin suggested a modification where vorticity near a flow boundary is represented on a basis of planar, tile-like elements. Many subsequent researchers have adopted these forms for investigations into two-dimensional flow.
For three-dimensional studies, an early proposal for the geometrical form of the three-dimensional elements was a collection of point vortices. As shown in FIG. 1, an incoming three-dimensional fluid flow, represented by arrow 10, is moving over a submerged body surface 11. Numerous point vortices 12, are chosen for computing vorticity (i.e. velocity) of the fluid flow 10. The spinning, represented by rotational arrows 13, of the fluid associated with each point vortex 12 is extremely fast (mathematically infinite) near the point itself. Accordingly, this approach requires an artificial smoothing or smearing out of point vortices 12 which leads to inaccurate vorticity distributions (i.e. velocity determination). In addition, numerous point vortices 12 are required to adequately represent the three-dimensional physical distribution of vorticity in the flow thereby leading to complex computations.